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We investigate the properties of the stochastic Gross-Pitaevskii equationdescribing a condensate interacting with a stationary thermal cloud derived byGardiner and coworkers. We find the appropriate Ehrenfest relations for theSGPE, including the effect of growth noise and projector terms arising from theenergy cutoff. This is carried out in the high temperature regime appropriatefor the SGPE, which simplifies the action of the projectors. The validitycondition for neglecting the projector terms in the Ehrenfest relations isfound to be more stringent than the usual condition of validity of thetruncated Wigner method or classical field method -- which is that all modesare highly occupied. In addition it is required that the overlap of thenonlinear term with the lowest energy eigenstate of the non-condensate band issmall. We show how to use the Ehrenfest relations along with the correctionsgenerated by the projector to monitor dynamical artifacts arising from thecutoff. We also investigate the effect of using different bases to describe aharmonically trapped BEC at finite temperature by comparing the condensatefraction found using the plane wave and single particle bases. We show that theequilibrium properties are strongly dependent on the choice of basis. There isthus an optimal choice of plane wave basis for a given cut-off energy and weshow that this basis gives the best reproduction of the single particlespectrum, the condensate fraction and the position and momentum densities.Comment: 23 pages, 5 figure

Properties of the stochastic Gross–Pitaevskii equation: finite temperature Ehrenfest relations and the optimal plane wave representation